Stephens' constant

Stephens' constant expresses the density of certain subsets of the prime numbers. Let $a$ and $b$ be two multiplicatively independent integers, that is, $a^m{b}^n\ne 1$ except when both $m$ and $n$ equal zero. Consider the set $T(a,b)$ of prime numbers $p$ such that $p$ evenly divides $a^k-b$ for some power $k$. Assuming the validity of the generalized Riemann hypothesis, the density of the set $T(a,b)$ relative to the set of all primes is a rational multiple of

$C_S=\prod _p(1-\frac{p}{p^3-1})=0.57595996889294543964316337549249669\dots$(sequence A065478 in the OEIS)

Stephens' constant is closely related to the Artin constant $C_A$ that arises in the study of primitive roots.

$C_S=\prod _p(C_A+\left(\frac{1-p^2}{p^2(p-1)}\right))\left(\frac{p}{(p+1+\frac{1}{p})}\right)$

See also

• Euler product
• Twin prime constant